Taking into account various discretizations and approximations of forward models to achieve computational speedups has a long tradition in scientific computing, e.g., multigrid solvers [1]}, [2]}, sparse grid approximations [3]}, multilevel Monte Carlo for estimating statistics [4]}, [5]}, [6]}; and multifidelity methods that leverage low-fidelity models without clear hierarchies [7]}. In terms of sampling from distributions, there is work on Markov chain Monte Carlo (MCMC) methods that exploit hierarchies of distributions such as
multistage MCMC methods [8]}, [9]},
multilevel Metropolis–Hastings [10]}; and MCMC methods with importance sampling [11]}.
Then, there are multilevel/multifidelity variational methods, where a transport map (flow) is parametrized a priori; for example, [12]}, [13]} build on [14]}, [15]} and construct the transport maps from a distribution on a lower level and then use it as proposal for Metropolis-Hastings or for importance sampling.
There are multilevel particle filters [16]} and multilevel sequential Monte Carlo [17]} methods, ensemble Kalman filtering [18]},
and extensions to nonlinear filtering using transport [19]}; these rely on telescoping sums of correlated differences between successive levels, whereas our approach uses the successive levels as preconditioners for sampling. Probably closest in style to our approach are the multilevel sequential Monte Carlo method [20]}, [21]} and the multilevel cross-entropy method [22]} that use distributions obtained on lower levels as starting distributions on higher levels.
